Recall that by the work of Eliashberg \(\xi _{\mathrm {st}}\)  is the unique tight contact structure on \(S^3\)  [1]} and that the overtwisted contact structures are in one-to-one correspondence to the homotopy classes of tangential 2-plane fields [2]}, which are on homology spheres in bijection with the integers via their normalized \(d_3\) -invariants [3]}. We denote the unique overtwisted contact structure on \(S^3\)  with normalized \(d_3\) -invariant equal to \(n\)  by \(\xi _n\) .
Contact surgery diagrams for all contact structures on \(S^3\)  where explicitly described in [4]}. A contact surgery diagram of \(\xi _1\)  is given by the contact \((+1)\) -surgery along the Legendrian unknot with Thurston–Bennequin invariant \(t=-2\)  and rotation number \(r=1\)  shown on the right of Figure REF . The contact \((\pm 1)\) -surgery diagram along the 2-component link shown on the left of Figure REF  represents \(\xi _{-1}\) . The disjoint union of two surgery diagrams describes a connected sum of the underlying contact manifolds. Since the \(d_3\) -invariant behaves additively under the connected sum, we get contact surgery diagrams of all contact structures on \(S^3\)  by taking appropriate disjoint unions of the contact surgery diagrams of \(\xi _1\)  and \(\xi _{-1}\) .
It follows that we can get \((S^3,\xi _1)\)  by a single contact \((+1)\) -surgery from \((S^3,\xi _{\mathrm {st}})\)  and that there exists a contact \((+1)\) -surgery on \((S^3,\xi _k)\)  yielding \((S^3,\xi _{k+1})\)  and thus conversely a contact \((-1)\) -surgery from \((S^3,\xi _k)\)  to \((S^3,\xi _{k-1})\) .
